Written by the Fiveable Content Team • Last updated September 2025
Written by the Fiveable Content Team • Last updated September 2025
Definition
Hanging cables, or catenaries, are curves formed by a cable suspended under its own weight and subject to hyperbolic functions. The shape of these cables can be described mathematically using the hyperbolic cosine function $\cosh(x)$.
5 Must Know Facts For Your Next Test
The equation for a hanging cable is $y = a \cosh(\frac{x}{a})$, where $a$ is a constant that depends on the physical properties of the cable.
The curve described by a hanging cable is called a catenary.
$\cosh(x)$ and $\sinh(x)$ are essential hyperbolic functions used to describe hanging cables.
Hanging cables provide real-world applications for integration techniques in calculus.
The hyperbolic cosine function $\cosh(x)$ has properties similar to those of the trigonometric cosine function but applies to different types of problems.
A curve formed by a perfectly flexible chain suspended between two points under its own weight, described mathematically by $y = a \cosh(\frac{x}{a})$.
$\cosh(x)$: $\cosh(x)$, or hyperbolic cosine, is defined as $\frac{e^x + e^{-x}}{2}$ and describes the shape of catenaries.
$\sinh(x)$: $\sinh(x)$, or hyperbolic sine, is defined as $\frac{e^x - e^{-x}}{2}$ and is often used alongside $\cosh(x)$ in problems involving hyperbolic functions.